Taylor and Y.-N

Using a one-way Monte Carlo algorithm from several different starting points, we get an approximation to the distribution of toric threefold bases that can be used in four-dimensional F-theory compactification. We separate the threefold bases into "resolvable" ones where the Weierstrass polynomials $(f,g)$ can vanish to order (4,6) or higher on codimension-two loci and the "good" bases where these (4,6) loci are not allowed. A simple estimate suggests that the number of distinct resolvable base geometries exceeds $10^{3000}$, with over $10^{250}$ "good" bases, though the actual numbers are likely much larger. We find that the good bases are concentrated at specific "end points" with special isolated values of $h^{1,1}$ that are bigger than 1,000. These end point bases give Calabi-Yau fourfolds with specific Hodge numbers mirror to elliptic fibrations over simple threefolds. The non-Higgsable gauge groups on the end point bases are almost entirely made of products of $E_8$, $F_4$, $G_2$ and SU(2). Nonetheless, we find a large class of good bases with a single non-Higgsable SU(3). Moreover, by randomly contracting the end point bases, we find many resolvable bases with $h^{1,1}(B)\sim 50-200$ that cannot be contracted to another smooth threefold base.